On Arthur packets containing a fixed tempered representation

1. Problem Statement

The central problem addressed in this paper is Problem 1.1: Given an irreducible tempered representation $\pi$ of a classical $p$-adic group $G_n$ (specifically split symplectic, odd special orthogonal, or even orthogonal groups), determine the number of local Arthur packets that contain $\pi$.

• Context: Local Arthur packets ($\Pi_\psi$) are finite multisets of representations associated with Arthur parameters $\psi$. While the Local Langlands Correspondence parameterizes tempered representations via bounded Arthur parameters, non-tempered parameters can also contain tempered representations.
• Challenge: Local Arthur packets often have nontrivial intersections. A single tempered representation $\pi$ can belong to multiple distinct Arthur packets. Previous combinatorial algorithms (e.g., by Mœglin, Atobe) could construct these packets but did not provide a uniform, explicit formula for counting how many packets contain a specific $\pi$.
• Significance: Solving this is crucial for understanding the structure of the local Langlands correspondence and has applications in the Gan-Gross-Prasad conjectures and theta correspondence (specifically regarding non-relevant Arthur parameters).

2. Methodology and Framework

The authors utilize the combinatorial framework of extended multi-segments to parameterize local Arthur packets. The methodology proceeds through several key reductions:

A. Reduction to Good Parity and Extended Multi-Segments

• Good Parity: Using Mœglin's reduction, the problem is reduced to the "good parity" case, where the Arthur parameter $\psi$ is self-dual and of the same type as the group.
• Parameterization: Tempered representations correspond to extended multi-segments $E$ where the support consists of singletons (integers) and the "length" parameters $l_i = 0$.
• Intersection Theory: The set of Arthur packets containing a fixed $\pi$ corresponds to the set of extended multi-segments $E'$ such that $\pi(E') = \pi(E)$. These segments are related by raising operators (combinations of row exchanges, union-intersections, and dual operators).

B. Decomposition into Blocks

A critical structural insight is Lemma 3.4: Any tempered extended multi-segment $E$ supported in the integers can be uniquely decomposed into disjoint blocks ($B_1, \dots, B_k$).

• A block is a maximal sub-segment satisfying specific alternating sign and support conditions.
• The decomposition is ordered, and the interaction between blocks is constrained by a "staircase property" regarding their supports.

C. Classification of Virtual Extended Multi-Segments (Type YM)

The authors introduce a class of virtual extended multi-segments of Type YM (Definition 5.12) to classify all segments equivalent to a given block.

• These are constructed using S-data (a collection of subsets of integers) and, for blocks starting at zero, T-data (partitions of those subsets).
• Theorem 5.17 establishes that the set of all Arthur parameters containing a block $B$ corresponds precisely to the set of Type YM segments with the same sign as $B$.

D. Operator Analysis

The authors analyze the action of raising operators (specifically $S, M, U, D$) on these Type YM segments:

• $S$ (Separation): Splits a chain of circles.
• $M$ (Merging): Merges two consecutive "hats" (segments with negative endpoints).
• $U$ (Un-merging): The inverse of merging, effectively separating a hat.
• $D$ (Dualizing): Applies a dual operator to a hat, converting it into a chain or merging it with a multiple.
• Independence of Blocks: A major technical achievement is proving (Lemma 7.12) that raising operators acting on one block do not affect the validity or structure of operations on other blocks in the decomposition. This allows the counting problem to be factored.

3. Key Contributions and Results

Theorem 3.6: Counting for Individual Blocks

The paper provides a recursive formula for the number of local Arthur packets containing a single block $B$, denoted $|\Psi(\pi(B))|$. The formula depends on whether the block starts at column 0 ($c_{min}=0$) or not, and the multiplicity of the column immediately preceding the maximum column ($m_{c_{max}-1}$).

• Case 1: $c_{min} = 0$ (Starts at zero)

  • If $m_{c_{max}-1} = 1$: $|\Psi(\pi(B))| = 3 \cdot |\Psi(\pi(B'))|$
  • If $m_{c_{max}-1} \in \{3, 5, \dots\}$: $|\Psi(\pi(B))| = 4 \cdot |\Psi(\pi(B'))| - |\Psi(\pi(B''))|$
  • (Where $B'$ and $B''$ are blocks with the last column removed).

• Case 2: $c_{min} > 0$ (Does not start at zero)

  • If $m_{c_{max}-1} = 1$: $|\Psi(\pi(B))| = 2 \cdot |\Psi(\pi(B'))|$
  • If $m_{c_{max}-1} \in \{3, 5, \dots\}$: $|\Psi(\pi(B))| = 3 \cdot |\Psi(\pi(B'))| - |\Psi(\pi(B''))|$

Theorem 3.7: Counting for General Tempered Representations

The main result extends the block count to the entire representation. If a tempered representation $\pi$ corresponds to an extended multi-segment $E$ decomposed into blocks $B_1, \dots, B_k$:
$$|\Psi(\pi(E))| = |\Psi(\pi(B_1))| \cdot \prod_{i=2}^k |\Psi(\pi(\text{sh}_1(B_i)))|$$

• Here, $\text{sh}_1(B_i)$ denotes the block $B_i$ shifted by 1 (effectively treating it as a block not starting at zero).
• This formula implies that the total count is the product of the counts of the individual blocks, with the exception of the first block which retains its "starting at zero" properties.

4. Significance and Implications

1. Explicit Combinatorial Formula: The paper resolves Problem 1.1 for the "integer case" (tempered representations whose associated extended multi-segments are supported entirely in the integers). It provides a concrete, recursive algorithm to compute the number of containing packets, moving beyond abstract existence proofs.
2. Generalization of Mœglin's Results: The result generalizes Mœglin's finding that supercuspidal representations lie in the maximum possible number of packets. The authors show that for general tempered representations, the count is determined by the specific combinatorial structure of their block decomposition.
3. Foundation for Theta Correspondence: The authors note in the introduction and Remark 6.3 that these results are instrumental in studying the theta lift to the first occurrence in the "going-up tower." This application is explored in a forthcoming article [10], linking the counting of Arthur packets to the local theta correspondence.
4. Partial Solution to the Half-Integer Case: While the paper solves the integer case completely, it identifies the "half-integer" case (where supports are in $\mathbb{Z} + 1/2$) as the remaining hurdle for a complete solution to Problem 1.1. The authors argue that the operators and combinatorial logic developed here will likely extend to that case, though with additional complexities (such as the partial dual operator).

Conclusion

This paper establishes a rigorous combinatorial framework for counting local Arthur packets containing a fixed tempered representation. By decomposing representations into independent blocks and analyzing the action of raising operators on "Type YM" virtual segments, the authors derive a recursive formula that completely solves the counting problem for the integer-supported case. This work bridges deep representation-theoretic questions with precise combinatorial algorithms, offering new tools for the Langlands program.